Prove that if in a triangle ABC the sides AB and AC are not equal, then the median AM of the triangle is not a height.

Let us prove by contradiction.
Let the median AM be the height of the triangle ABC.
Then triangles ABM and AFM are rectangular.
Moreover, the hypotenuses AB and AC are not equal in terms of the condition, and one of the legs is equal to the other.
This is the common side of AM.
Consequently, the other legs BM and MC cannot be equal, since if the legs are equal in the triangles, then the hypotenuses must also be equal, but this is not the case.
But, since AM and median by condition, BM and MS should be equal.
Hence it follows that AM is not a height.